Categorical Langlands correspondence for $\operatorname{SO}_{n,1}(\mathbb{R})$

In the context of the local Langlands philosopy for $\mathbb{R}$, Adams, Barbasch and Vogan describe a bijection between the simple Harish-Chandra modules for a real reductive group $G(\mathbb{R})$ and the space of ``complete geometric parameters''--a space of equivariant local systems on a variety on which the Langlands-dual of $G(\mathbb{R})$ acts. By a conjecture of Soergel, this bijection can be enhanced to an equivalence of categories. In this article, that conjecture is proven in the case where $G(\mathbb{R})$ is a generalized Lorentz group $\operatorname{SO}_{n,1}(\mathbb{R})$.